Carolyn Brown and Graham Hutton. Categories, allegories and circuit design. In LICS 9, pages 372-381, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a monoid with antiinvolution plus an operation with axioms resembling the intersection in sets, and a group of axioms very geometrical in flavor. Recent indications of the practical significance of this theory for computer science are the work on hardware component allegories [7] circuit design [3], and algebra of programming [2] There seems to be no algorithmic study of the pure equational theory of allegories (by pure we mean the theory of allegories as defined in page 196 in [6] without any further axioms like tabular, representable, unitary, etc. Freyd and Scedrov showed in [6] that ....

C. Brown and G. Hutton. Categories, Allegories and Circuit Design. Proceedings of the 9th Symposium on Logic in Computer Science, IEEE, 1994.

....essentially (a more concrete version of) the term graphs we are going to introduce. Let us mention also that, in di erent research areas, some authors have elaborated on the observation that (suitable kinds of) diagrams can be used to denote relations and to calculate on them: see, among others, [3,14,25]. Therefore on the one hand the use of sharing when reasoning about nondeterministic operations is quite well understood; on the other hand many other approaches stick to standard terms and address the problems sketched above directly in the design of the logical system. As a typical example, ....

C. Brown and G. Hutton. Categories, allegories and circuit design. In Logic in Computer Science, pages 372-381. IEEE Computer Society Press, 1994.

....of (suitable kinds of) diagrams to denote relations and to calculate on them, is witnessed not only by the use of sharing in papers like [AC79,Hes88,KB99] as discussed in Sect. 3, but also, for example, by the graphical calculi of [CL95,Kah97] and by applications to circuit design as in [BH94] 10 Given a multi algebra A : TG Rel, it is not dicult to present the set valued function associated with a given term graph: we show this by an example. Let us consider graph G 3 of the drawing after Def. 6. Then G A 3 is a function from A(1) 2 to P(A(1) and it is de ned as a 2 G ....

C. Brown and G. Hutton. Categories, Allegories and Circuit Design. In Proc. 9th Annual Symposium on Logic in Computer Science, Paris, France, pages 372-381. IEEE Computer Society Press, 1994.

.... was central to early investigations of logic and the foundations of mathematics [10, 18, 22, 23, 24] and has more recently found application in program speci cation and derivation, 2, 6, 4, 16] denotational and axiomatic semantics of programs, 8, 9, 20, 17] and hardware design and veri cation [7, 15]. The collection of binary relations on a set has rich algebraic structure: it forms a monoid under composition, each relation has a converse, and it forms a Boolean algebra under the usual set theoretic operations. In fact the equational theory in this language is undecidable, since it is ....

....derives from the idea of taking certain graphs as the representation of relations. These graphs, called here diagrams, arise very naturally and have been used since Peirce by researchers in the relation community (e.g. Tarski, Lyndon, J onsson, Maddux, etc. recent formalizations appear in [11, 1, 7]. What we do here is to take graphs seriously as a notation alternative to rst order terms, i.e. to treat diagrams as rst class syntactic entities, and speci cally as candidates for rewriting. One can see diagram rewriting as an instance of a standard technique in automated deduction. It is ....

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C. Brown and G. Hutton. Categories, allegories and circuit design. In Logic in Computer Science, pages 372-381. IEEE Computer Society Press, 1994.

....oe R P P P P Pi outl P P P P Pq outr oe S i i i i i) outr = This yields the pictorially intuitive idea of products being represented as parallel arrows; the graphical representation makes it easier to reason about each element of the pair separately. 3.4. 4 Related work Brown and Hutton [17] have developed a calculus of pictures, oriented towards circuit design. Their pictures are built up from basic cells and wires using sequential composition, intersection and reciprocation. They give a semantics to pictures in terms of relations, in a manner very similar to our approach. In [17, ....

....[17] have developed a calculus of pictures, oriented towards circuit design. Their pictures are built up from basic cells and wires using sequential composition, intersection and reciprocation. They give a semantics to pictures in terms of relations, in a manner very similar to our approach. In [17, 18] it is shown that their calculus is complete in that two pictures are equivalent with respect to their transformation rules if and only if they represent the same relation for all interpretations of the basic cells; this proof proceeds by viewing pictures as arrows in a unitary pretabular allegory ....

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Carolyn Brown and Graham Hutton. Categories, allegories and circuit design. In Ninth Annual IEEE Symposium on Logic In Computer Science, pages 372--381, 1994.

....mathematical formulae for example relations or sequential relations by graphs. We have presented rules for transforming graphs and explained how these rules affect the corresponding formulae. In this final section we discuss a few other points of interest. 4. 1 Related work Brown and Hutton [2] have developed a calculus of pictures, oriented towards circuit design. Their pictures are built up from basic cells and wires using sequential composition, intersection and reciprocation. They give a semantics to pictures in terms of relations, in a manner very similar to our approach. In [2, 3] ....

....[2] have developed a calculus of pictures, oriented towards circuit design. Their pictures are built up from basic cells and wires using sequential composition, intersection and reciprocation. They give a semantics to pictures in terms of relations, in a manner very similar to our approach. In [2, 3] it is shown that their calculus is complete in that two pictures are equivalent with respect to their transformation rules if and only if they represent the same relation for all interpretations of the basic cells; this proof proceeds by viewing pictures as arrows in a unitary pretabular allegory ....

Carolyn Brown and Graham Hutton. Categories, allegories and circuit design. In Ninth Annual IEEE Symposium on Logic in Computer Science, pages 372--381, 1994.

....to employ both methods extensively, and, independently of other approaches, have been driven to develop a graphical calculus for making complex relation algebraic proofs more accessible. It turns out that, although our approach shares many common points with those presented in the literature [BH94, CL95], it still is more general and more flexible than those approaches since we draw heavily on additional background in algebraic graph rewriting (see [EKL90] for a tutorial overview) The part of the structure of relation algebra that can readily be exploited in graphical calculi is that of a ....

....unitary pretabular allegory (UPA, introduced in [FS90] Allegories are a generalisation of categories to cope with relation like structures; we shall not need any allegory theory in this paper, but only refer to it for comparison with one of the main streams of related work in the literature. In [BH94], an approach to transformations of expressions in UPAs via transformations of graphs has been presented and proven correct. The approach has been developed with a bias towards VLSI circuit development and the formalisation and drawings reflect this. More or less building on the approach of ....

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Carolyn Brown and Graham Hutton. Categories, allegories and circuit design. In Proceedings, Ninth Annual IEEE Symposium on Logic in Computer Science, pages 372--381, Paris, France, 4--7 July 1994. IEEE Computer Society Press.

....mathematical formulae for example relations or sequential relations by graphs. We have presented rules for transforming graphs and explained how these rules affect the corresponding formulae. In this final section we discuss a few other points of interest. 4. 1 Related work Brown and Hutton [2] have developed a calculus of pictures, oriented towards circuit design. Their pictures are built up from basic cells and wires using sequential composition, intersection and reciprocation. They give a semantics to pictures in terms of relations, in a manner very similar to our approach. In [2,3] ....

....[2] have developed a calculus of pictures, oriented towards circuit design. Their pictures are built up from basic cells and wires using sequential composition, intersection and reciprocation. They give a semantics to pictures in terms of relations, in a manner very similar to our approach. In [2,3] it is shown that their calculus is complete in that two pictures are equivalent with respect to their transformation rules if and only if they rep17 resent the same relation for all interpretations of the basic cells; this proof proceeds by viewing pictures as arrows in a unitary pretabular ....

Carolyn Brown and Graham Hutton. Categories, allegories and circuit design. In Ninth Annual IEEE Symposium on Logic in Computer Science, pages 372-- 381, 1994.

....language Ruby [6] are used to derive hardware circuits from abstract specifications of their behaviour. Freyd and Scedrov [3] have recently introduced allegories, which generalise the notion of sets and relations in the same sense that categories generalise the notion of sets and functions. In [1], Brown and Hutton showed how allegories could be used to reason about a pictorial representation of Ruby programs. In this paper, we develop the mathematical notions used in [1] and give sketch proofs of expressiveness, normalisation and completeness. Our results provide new methods of formal ....

....the notion of sets and relations in the same sense that categories generalise the notion of sets and functions. In [1] Brown and Hutton showed how allegories could be used to reason about a pictorial representation of Ruby programs. In this paper, we develop the mathematical notions used in [1], and give sketch proofs of expressiveness, normalisation and completeness. Our results provide new methods of formal reasoning about relational programs. We define a syntax for hardware circuit design, based on unitary pretabular allegories, which are allegories with a certain product structure. ....

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Carolyn Brown and Graham Hutton. Categories, allegories and circuit design. To appear in Proc. LICS, 1994.

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Carolyn Brown and Graham Hutton. Categories, allegories and circuit design. In LICS 9, pages 372-381, 1994.

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C. Brown and G. Hutton. Categories, allegories and circuit design. In Logic in Computer Science, pages 372-381. IEEE Computer Society Press, 1994.

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